In the 1970s, Almkvist and Fossum gave formulae which describe completely the decomposition of symmetric powers of modular representations of cyclic groups into indecomposable summands. We show how (in spite of the wildness of the representation type) some of their results can be generalized to representations of elementary abelian p-groups. Some applications to invariant theory will also be given.

06/10/2014 5:30 PM

103

Alonso Castillo Ramirez (Zaragoza)

Majorana algebras

A Majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some properties of the 2A-axes of the 196884-dimensional Monster Griess algebra. The term was introduced by A. A. Ivanov in 2009 inspired by Sakuma's and Miyamoto's work on vertex operator algebras. In this talk, we are going to present some elementary examples of Majorana algebras, and we will sketch how to obtain the automorphism groups and maximal associative subalgebras of the two-generated Majorana algebras.

13/10/2014 5:30 PM

103

Melanie de Boeck (Kent)

Studying Foulkes modules using semistandard homomorphisms

The action of the symmetric group Smnon set partitions of a set of size mn into n sets of size m gives rise to a permutation module called the Foulkes module. Structurally, very little is known about Foulkes modules, even in characteristic zero. In this talk, we will see that semistandard homomorphisms may be used as a tool for studying the module structure and, in particular, for establishing relationships between irreducible constituents of Foulkes modules.

20/10/2014 6:00 PM

103

Anton Evseev (Birmingham)

Graded RoCK blocks and wreath products

The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic p, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block B0of the wreath product Spwr Sw, where w is the "weight" of the block. More precisely (and more simply), the conjecture states that the idempotent truncation in question is isomorphic to a tensor product of B0and a certain matrix algebra. The talk will outline a proof of this conjecture, which uses an isomorphism between the group algebra of a symmetric group and a cyclotomic Khovanov-Lauda-Rouquier algebra and the resulting grading on the group algebra of the symmetric group. This result generalizes a theorem of Chuang-Kessar, which applies to the case w

27/10/2014 4:30 PM

103

Maria Calvo Cervera (Granada)

Cohomological classification of monoidal groupoids

Strongly inspired by Schreier’s analysis of group extensions and its extension to fibrations of categories by Grothendieck, we analyse the structure of monoidal categories in which every arrow is invertible. In particular, we state precise classification theorems for those monoidal groupoids whose isotropy groups are all abelian, by means of Leech’s cohomology groups of monoids.

03/11/2014 4:30 PM

No seminar - School colloquium

10/11/2014 4:30 PM

103

Emilio Pierro (Birkbeck)

The Möbius function of the small Ree groups

In 1936 Hall showed that Möbius inversion could be applied to the lattice of subgroups of a finite group G in order to determine the number of n-bases of G, that is, generating sets of G of size n. The question can be modified and n-bases subject to certain relations can also be enumerated with applications to the theory of Riemann surfaces, Hurwitz groups, dessins d'enfants and various other algebraic, topological and combinatorial enumerations. In order to determine the Möbius function of a group it is necessary to understand the subgroup structure of a group and so we also give a description of the simple small Ree groups R(q)=2G2(q), in particular their maximal subgroups, in terms of their 2-transitive permutation representations of degree q3+1.

17/11/2014 4:30 PM

No seminar - School colloquium

24/11/2014 4:30 PM

103

Keith Brown (UEA)

Properly stratified quotients of Khovanov-Lauda-Rouquier algebras

Introduced in 2008 by Khovanov and Lauda, and independently by Rouquier, the KLR algebras are a family of infinite-dimensional graded algebras which categorify the negative part of the quantum group associated to a graph. In finite types these algebras are known to have nice homological properties, in particular they are affine quasi-hereditary. In this talk I'll explain what it means to be affine quasi-hereditary and how this relates to properties of finite dimensional algebras. I'll then introduce a finite dimensional quotient of the KLR algebra which preserves some of the homological structure of the original algebra and provide a bound on its finitistic dimension. This work will form part of my PhD thesis, supervised by Dr Vanessa Miemietz.

01/12/2014 4:30 PM

No seminar - School colloquium

08/12/2013 4:30 PM

103

Adam Boocher (Edinburgh)

Ideals of maximal minors and their resolutions

The set of k by n matrices with rank at most r naturally forms an algebraic variety. Its defining equations are given by determinants and it enjoys many beautiful properties. In this talk I'll discuss some recent work that describes how this variety behaves upon specialization with some applications to matroids and free resolutions.

09/01/2015 4:30 PM

Engineering 371

Xin Li (QMUL)

On subsemigroups of groups

We discuss algebraic properties of subsemigroups of groups. These properties first arose in the study of operator algebras attached to semigroups. In this talk, however, the focus will not be on operator algebras. On the one hand, this means that no operator algebraic prerequisites are required, and on the other hand, it allows us to focus on purely algebraic aspects which are hopefully interesting on their own right. Our main concepts will be illustrated by natural examples like Braid groups, Artin groups or the Thompson group.

02/02/2015 4:30 PM

103

Thomas Kahle (OvGU Magdeburg)

How primary decomposition of monoid congruences and binomial ideals is wrong

In a polynomial ring, a binomial says that two monomials are scalar multiples of each other. Forgetting about the scalars, a binomial ideal describes an equivalence relation on the monoid of exponents. Ideally one would want to carry out algebraic computations, such as primary decomposition of binomial ideals, entirely in this combinatorial language. We will present such a calculus, enabling one to compute by looking at pictures of monoids.

The RSK correspondence assigns a pair of standard tableaux to every element of the symmetric group. This describes a partitioning of the group into “cells”. More generally these cells can be defined for any Coxeter group. Recently Henriques and Kamnitzer defined an action of the “cactus group” on crystals for semisimple Lie algebras. I will explain, in type A, the connection between this action and a conjectural method of Bonnafe and Rouquier of defining cells for the symmetric group. I will show how this action appears using Schubert calculus or alternatively using the representation theory of the symmetric group and certain generalisations of the Jucys-Murphy elements called the Gaudin Hamiltonians.

Specht modules play an integral role in the representation theory of the symmetric groups. Recent work by Brundan and Kleshchev and Khovanov, Lauda and Rouquier has added a wealth of structure to the Specht modules in positive characteristic. One ambitiously hopes to obtain a graded analogue of the hook length formula, introduced by Frame, Robinson and Thrall in 1954, which calculates the dimension of the Specht modules.

I will begin with the combinatorial construction of the Specht modules over a field of characteristic 0, as first developed by G. D. James in the 1970s. I will then give a review of the recent developments in modular representation theory of the symmetric groups, together with my progress in attaining a graded dimension formula for the Specht modules.

Classical Tannaka duality is a duality between groups and their categories of representations. It answers two basic questions: can we recover the group from its category of representations, and can we characterize categories of representations abstractly? These are often called the reconstruction problem and the recognition problem. In the context of affine group schemes over a field, the recognition problem was solved by Saavedra and Deligne using the notion of a (neutral) Tannakian category.

In this talk I will explain how this theory can be generalized to the context of certain algebraic stacks and their categories of coherent sheaves (using the notion of a weakly Tannakian category). On Tuesday [in the Quantum Algebras seminar] I will talk about work in progress to construct universal weakly Tannakian categories and some of their applications. The aim is to interpret various constructions on stacks (for example fiber products) in terms of the corresponding weakly Tannakian categories.

16/03/2015 4:30 PM

103

Arkady Vaintrob (Oregon)

Cohomological field theories related to singularities and matrix factorizations

I will discuss a cohomological field theory associated to a quasihomogeneous isolated singularity W with a group G of its diagonal symmetries. The state space of this theory is the equivariant Milnor ring of W and the corresponding invariants can be viewed as analogs of the Gromov-Witten invariants for the non-commutative space associated with the pair (W,G). In the case of simple singularities of type A they control the intersection theory on the moduli space of higher spin curves. The construction is based on derived categories of (equivariant) matrix factorizations of W.

We will describe combinatorial "models" that can be used to study various quantum algebras (for example quantum matrices, quantum symmetric and skew-symmetric matrices, the quantum grassmannian and more). For all of these algebras, there is an action of an algebraic torus by automorphisms and a description of the torus-invariant prime ideals is a key step towards understanding the full prime spectrum due to work of Goodearl and Letzter. We will discuss how the above combinatorial models can be used to calculate Grobner bases of all torus-invariant prime ideals, as well as provide other useful information. Portions of this talk are joint work with Stephane Launois.

The talk will begin with an introduction to difference algebraic groups, i.e., groups defined by algebraic difference equations. Like étale algebraic groups can be described as finite groups with a continuous action of the absolute Galois group of the base field, étale difference algebraic groups can be described as certain profinite groups with some extra structure. Étale difference algebraic groups satisfy a decomposition theorem that shows that they can all be build from étale algebraic groups and finite groups equipped with an endomorphism.

Lie groups&algebras, Coxeter/reflection groups and root systems are closely related, and feature prominently throughout mathematics and physics, in particular the exceptional ones. We argue that the root system concept is the most useful for our purposes, and that since an inner product is implicit when considering reflections, one can always construct the Clifford algebra over the underlying vector space. Clifford algebra has a very simple reflection formula and via the Cartan-Dieudonne theorem provides a double cover of the orthogonal transformations. In particular, in 3D the Clifford algebra is 8-dimensional and its even subalgebra is 4-dimensional. Starting from a 3D root system one can therefore construct groups of 4D or 8D objects under Clifford multiplication. The 4D ones can in general be shown to be root systems with interesting automorphism groups - in particular D4, F4, H4 are induced from A3, B3, H3 - and for the 8D case one can show (via R. Wilson's reduced inner product) that the Clifford double cover of the 120 reflections in H3 yields the 240 roots of E8.

25/01/2016 4:30 PM

(No seminar – School Colloquium)

15/02/2016 4:30 PM

(No seminar – School Colloquium)

29/02/2016 4:30 PM

103

Rob Wilson (QMUL)

[Cancelled]

*Clifford algebra or Lie algebra: what are the Dirac matrices?*

The Dirac equation reconciles quantum mechanics with special relativity, by describing the wave functions of particles like the electron travelling at relativistic speeds. It is a PDE with coefficients which are 4 x 4 complex matrices called the Dirac gamma matrices. Conventional wisdom states that these matrices generate the Clifford algebra Cl(3,1) for the quadratic form with signature (3,1). However, their use in physics requires multiplying some of the Clifford algebra by i, thereby destroying the Clifford algebra structure. I argue that it makes more sense to say the gamma matrices generate the Lie algebra so(5,1). This viewpoint potentially throws light on the nature of the weak force, and thereby on the nature of mass and charge.

I will start by presenting the theory of the spaces spanned by the local pieces of thesepiecewise (quasi-)polynomial functions and point out connections with matroid theory.This theory has been developed in the 1980s by Dahmen and Micchelli. Later it has beenput in a broader context by De Concini, Procesi, Vergne and others.Then I will present a refined version of the Khovanskii-Pukhlikov formula that relates thevolume and the number of integer points of a smooth lattice polytope.

14/03/2016 4:30 PM

(No seminar – School Colloquium)

21/03/2016 4:30 PM

103

Imen Belmokhtar (QMUL)

The structure of induced simple modules of 0-Hecke algebras

In this talk we shall be concerned with the induced simple modules of the 0-Hecke algebras of types A and B.

The irreducible representations of 0-Hecke algebras were classified and shown to be one-dimensional by Norton in 1979.

To understand the structure of a finite-dimensional module, one would ideally like to know its full submodule lattice; this is easily computable for small dimensions but much harder for larger ones. Given certain conditions, a smaller poset encoding the submodule lattice can be rather easily obtained.

We shall discuss the theory allowing us to get this smaller poset and build on results by Fayers in the type A case to state new results in type B.

03/10/2016 5:00 PM

Maths LT

No seminar owing to School colloquium

10/10/2016 4:30 PM

FB 3.11

Leonard H. Soicher (QMUL)

Block intersection polynomials and strongly regular graphs

I will give a brief introduction to block intersection polynomials, andthen discuss their application to the study of strongly regular graphs,in particular describing recent joint work with Gary Greaves onnew upper bounds for the clique numbers of strongly regular graphsin terms of their parameters. No previous knowledge of stronglyregular graphs will be assumed.

17/10/2016 4:30 PM

FB 3.11

Robert A. Wilson (QMUL)

Principal ideal domains and Euclidean domains

24/10/2016 4:30 PM

FB 3.11

Wajid Mannan (QMUL)

Group presentations, representations over the integers and homotopy

31/10/2016 5:00 PM

Maths LT

No seminar owing to School colloquium

14/11/2016 4:30 PM

FB 3.11

Cecilia Busuioc (QMUL)

K-theory and Arithmetic

In this talk, I will give a brief account of the deep connection between the geometry of modular curves and the arithmetic of cyclotomic fields, originally conjectured by R. Sharifi.The main idea relies on a K-theoretic construction of modular symbols that enjoys furthergeneralisations to a GL_n -setting. This is the subject of a work in progress with G. Stevens and O. Patashnick.

21/11/2016 4:30 PM

FB 3.11

John N. Bray (QMUL)

Representations of some finite groups

28/11/2016 4:30 PM

FB 3.11

Tomasz Popiel (QMUL)

Symmetries of generalised polygons

Generalised polygons are point?line incidence geometries introduced by Jacques Tits in an attempt to find geometric models for finite simplegroups of Lie type. A famous theorem of Feit and G. Higman asserts that the only "non-trivial"examples are generalised triangles (projectiveplanes), quadrangles, hexagons and octagons. In each case, there are "classical" examples associated with certain Lie type groups, and in thelatter two cases these are the only known examples. The classical examples are highly symmetric; in particular, their automorphism groups acttransitively on flags and primitively on both points and lines. There have been various attempts to classify generalised polygons subject tosymmetry assumptions whether weaker, stronger, or just different to those mentioned above and perhaps one of the strongest results in thisdirection is a theorem of Kantor from 1987, asserting that a point-primitive projective plane is either classical (Desarguesian) or has aprime number of points and a severely restricted automorphism group. I will review some on-going work with John Bamberg, Stephen Glasby, LukeMorgan, Cheryl Praeger and Csaba Schneider that aims to classify the point-primitive generalised quadrangles, hexagons and octagons.

05/12/2016 5:00 PM

Maths LT

No seminar owing to School colloquium

08/05/2017 3:00 PM

W316

Roozbeh Hazrat (Western Sydney University)

Leavitt path algebras

From a directed graph one can generate various algebras that capture the movements along the graph. One such algebra is the Leavitt path algebra.

Despite being introduced only 10 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*-algebras, a theory which has become an area of intensive research globally. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered.

In this talk, we introduce Leavitt path algebras and try to classify them by means of (graded) Grothendieck groups. We will ask nice questions!

25/09/2017 4:30 PM

Queens W316

Steve Lester (QMUL)

Superscars for wave functions of a point scatterer on the torus

A fundamental problem in Quantum Chaos is to understand the distribution of mass of Laplace eigenfunctions on a given smooth Riemannian manifold in the limit as the eigenvalue tends to infinity. In this talk I will consider a Laplace operator perturbed by a delta potential (point scatterer) on the torus and describe the distribution of mass of the eigenfunctions of this operator. It turns out that in this setting, the distribution of mass of the eigenfunctions is related to properties of integers which are representable as sums of two squares. I will describe this relationship and indicate how tools from analytic number theory such as sieve methods and the theory of multiplicative functions can be used to study the relevant properties of such integers.

09/10/2017 4:30 PM

Queens W316

Tobias Berger (Sheffield)

Paramodularity of abelian surfaces

The key ingredient in Wiles' proof of Fermat's last theorem was to establish the modularity of elliptic curves. Despite many impressive advances in the Langlands programme the analogous question of modularity for abelian varieties of dimension 2 is far from settled. I will report on work in progress with Kris Klosin on the modularity of Galois representations $G_{\mathbf{Q}} \to {\rm GSp}_4(\mathbf{Q}_p)$ that are residually reducible. I will explain, in particular, how this can be used in certain cases to verify Brumer and Kramer's paramodular conjecture for abelian surfaces over Q with a rational torsion point of order p.

16/10/2017 4:30 PM

Queens W316

Vanessa Miemietz (UEA)

2-representations of finitary 2-categories

I will give an introduction to 2-representation theory and will give an overview of the state of the art for finitary 2-categories, which should be seen as 2-analogues of finite-dimensional algebras.

23/10/2017 4:30 PM

Queens 316

Peter Humphries (UCL)

The Conductor and the Newform for Representations of GL_n(R) and GL_n(C)

There is a well-known theory of decomposing spaces of automorphic forms into subspaces spanned by newforms and oldforms, and associated to a newform is its conductor. This theory can be reinterpreted as a local statement, and generalised to GL_n, as distinguishing certain vectors in a generic irreducible admissible representation of GL_n(F), where F is a nonarchimedean local field, and associating to this representation a conductor (or rather, a conductor exponent). Such a local theory was previously not well understood for archimedean fields. In this talk, I will introduce this theory in this hitherto unexplored setting.

13/11/2017 4:30 PM

Queens W316

Rachel Newton (Reading)

Counting failures of a local-global principle

The search for rational solutions to polynomial equations is ongoing for more than 4000 years. Modern approaches try to piece together 'local' information to decide whether a polynomial equation has a 'global' (i.e. rational) solution. I will describe this approach and its limitations, with the aim of quantifying how often the local-global method fails within families of polynomial equations arising from the norm map between fields, as seen in Galois theory. This is joint work with Tim Browning.

For any finite group G and any prime p, it is interesting to ask which ordinary irreducible representations remain irreducible modulo p. For the symmetric and alternating groups this problem was solved several years ago. Here we look at projective representations of symmetric groups, or equivalently representations of double covers of symmetric groups, focussing on characteristic 2 (which behaves very differently from odd characteristic). I'll give the classification of which irreducibles remain irreducible in characteristic 2, and describe some of the methods used in the proof. I'll assume some basic knowledge of representation theory, but I'll introduce projective representations and double covers from scratch.

04/12/2017 4:30 PM

Queens W316

Tomasz Popiel (QMUL)

TBA

26/01/2015 4:30 PM

103

Sarah Brodsky (TU Berlin)

Moduli of Tropical Plane Curves

Tropical curves have been studied under two perspectives; the first perspective defines a tropical curve in terms of the tropical semifieldT=(R∪{-∞}, max, +), and the second perspective defines a tropical curve as a metric graph with a particular weight function on its vertices. Joint work with Michael Joswig, Ralph Morrison, and Bernd Sturmfels, we study which metric graphs of genus g can be realized as smooth, plane tropical curves of genus g with the motivation of understanding where these two perspectives meet.

Using Polymake, TOPCOM, and other computational tools, we conduct our study by constructing a map taking smooth, plane tropical curves of genus g into the moduli space of metric graphs of genus g and studying the image of this map. In particular, we focus on the cases when g=2,3,4,5. In this talk, we will introduce tropical geometry, discuss the motivation for this study, our methodology, and our results.

The notion of a synchronizing permutation group arose from attempts to prove the long-standing Černý conjecture in automata theory. The class of synchronizing permutation groups is of interest in its own right, and lies strictly between the classes of finite primitive permutation groups and finite 2-transitive groups. I will discuss my recent determination of the synchronizing permutation groups of degree at most 255, using my newly developed algorithms and programs for proper vertex-k-colouring a graph making use of that graph's automorphism group.

This seminar may be of interest to combinatorialists as well as algebraists.

26/01/2015 4:30 PM

103

Sarah Brodsky (TU Berlin)

Moduli of Tropical Plane Curves

Tropical curveshave been studied under two perspectives; the first perspective defines a tropical curve in terms of thetropical semifieldT=(R∪{-∞}, max, +), and the second perspective defines a tropical curve as a metric graph with a particular weight function on its vertices. Joint work with Michael Joswig, Ralph Morrison, and Bernd Sturmfels, we study which metric graphs of genusgcan be realized as smooth, plane tropical curves of genusgwith the motivation of understanding where these two perspectives meet.

UsingPolymake,TOPCOM, and other computational tools, we conduct our study by constructing a map taking smooth, plane tropical curves of genusginto the moduli space of metric graphs of genus g and studying the image of this map. In particular, we focus on the cases wheng=2,3,4,5. In this talk, we will introduce tropical geometry, discuss the motivation for this study, our methodology, and our results.

02/03/2015 4:30 PM

No seminar ― School colloquium

01/02/2016 4:30 PM

103

Goran Malić (Manchester)

Maps on surfaces, matroids and Galois theory

Let M be a map on a connected, closed and orientable surface X. If B is a subset of the edge-set of M such that X\B is connected, then we say that B is a base of M. The collection of all bases of M form a delta-matroid, also known as a Lagrangian matroid. Analogously to matroids, there are two rich families of Lagrangian matroids: those that arise from embedded graphs, and those that arise from maximal isotropic subspaces of symplectic vector spaces.

Aside from the usual contraction and deletion operations, Lagrangian matroids admit twists; in the case of embedded graphs, twists of Lagrangian matroids correspond to the operation of partial duality, introduced by Chmutov in 2009. A partial dual of a map M is a map with only some of the edges dualised, and it can be interpreted as an intermediate step between M and its dual map M*.

In this talk I shall explain the relationship between maps, Lagrangian matroids, their twists, and partial duals. I shall also talk about a family of abstract tropical curves that arises from a map and its partial duals, and how it fits with the Galois-theoretic aspect of maps on surfaces (in the sense of Grothendieck's programme on dessins d'enfants).

08/02/2016 4:30 PM

103

Vincent Pilaud (CNRS & LIX, École Polytechnique)

Brick polytopes, lattice quotients, and Hopf algebras

This talk is motivated by the deep connections between the combinatorial properties of permutations, binary trees, and binary sequences. Namely, classical surjections from permutations to binary trees (BST insertion) and from binary trees to binary sequences (canopy) yield:∙ lattice morphisms from the weak order, via the Tamari lattice, to the boolean lattice;∙ normal fan coarsenings from the permutahedron, via Loday's associahedron, to the parallelepiped generated by the simple roots;∙ Hopf algebra inclusions from Malvenuto-Reutenauer's algebra, via Loday-Ronco's algebra, to Solomon's descent algebra.In this talk, we present an extension of this framework to acyclic k-triangulations of a convex (n+2k)-gon, or equivalently to acyclic pipe dreams for the permutation (1, …, k, n+k, …, k+1, n+k+1, …, n+2k). These objects are in bijection with the classes of the congruence of the weak order on S_n defined as the transitive closure of the rewriting rule U a c V_1 b_1 ⋯ V_k b_k W = U c a V_1 b_1 ⋯ V_k b_k W, for letters a < b_1, …, b_k < c and words U, V_1, …, V_k, W on [n]. It enables us to transport the known lattice and Hopf algebra structures from the congruence classes to these acyclic pipe dreams. We will describe the cover relations in this lattice and the product and coproduct of this algebra in terms of pipe dreams. We will also recall the connection to the geometry of the brick polytope.

22/02/2016 4:30 PM

103

Yankı Lekili (King's)

Koszul duality patterns in Floer theory

Abstract: We study symplectic invariants of the open symplectic manifolds X_Γ obtained by plumbingcotangent bundles of 2-spheres according to a plumbing tree Γ. For any tree Γ, we calculate(DG-)algebra models of the Fukaya category F(X_Γ) of closed exact Lagrangians in X_Γ and thewrapped Fukaya category W(X_Γ). When Γ is a Dynkin tree of type An or Dn (and conjecturallyalso for E6 , E7, E8 ), we prove that these models for the Fukaya category F(X_Γ) and W(X_Γ) arerelated by (derived) Koszul duality. As an application, we give explicit computations of symplecticcohomology of X_Γ for Γ = An, Dn , based on the Legendrian surgery formula. In thecase that Γ is non-Dynkin, we merely obtain a spectral sequence that converges to symplecticcohomology whose E2 -page is given by the Hochschild cohomology of the preprojective algebraassociated to the corresponding Γ. This is joint work with Tolga Etgü.

30/03/2015 5:30 PM

No seminar ― School colloquium

05/10/2015 5:30 PM

(No seminar – School Colloquium)

19/10/2015 5:30 PM

(No seminar – School Colloquium)

07/12/2015 4:30 PM

(No seminar – School Colloquium)

04/12/2017 4:30 PM

Queens W316

Tomasz Popiel (QMUL)

The symmetric representation of lines in PG(F^3 ⊗ F^3)

Tensors have numerous applications in areas such as complexity theory and data analysis, where it is often necessary to understand ‘decompositions’ and/or ‘canonical forms’ of tensors in certain tensor product spaces. Such problems are often studied over the complex numbers, but there are also reasons to to study them over finite fields, including connections with classifications of semifields. In this talk, I will discuss the following problem. Consider the vector space V of 3x3 matrices over a finite field F, i.e. the tensor product of F^3 with itself. The 1-dimensional subspaces spanned by the fundamental (or rank-1) tensors in V form the so-called Segre variety in the projective space PG(V), and the setwise stabiliser G in PGL(V) of this variety may be identified with PGL(3,F) acting via g in G taking a matrix representative A to g^TAg. The G-orbits of points and lines in the ambient projective space PG(V) were determined by Michel Lavrauw and John Sheekey (Linear Algebra Appl. 2015). I will discuss joint work with Michel Lavrauw in which we determine which of the G-line orbits can be represented by symmetric 3x3 matrices, i.e. we classify the orbits of lines in PG(V) under the setwise stabiliser K of the so-called Veronese variety. Interestingly, several of the G-line orbits that have such ‘symmetric representatives’ split under the action of K, and in many cases this splitting depends on the characteristic of F. Connections are also drawn with old work of Jordan, Dickson and Campbell on the classification of ternary quadratic forms.

08/10/2018 4:30 PM

Queens' Building, Room: W316

Charles R. Leedham-Green (QMUL)

Condorcet domains

A Condorcet domain of degree $d$ is a subset of the symmetric group of degree $d$ satisfying a condition that relates to the mathematics of choice. I have no interest in the mathematics of choice, but these objects turn out to have interesting properties.

The main challenge has been to find large Condorcet domains of given degree, and we have been using various techniques, from supercomputers to cardboard, with some theoretical ideas thrown in, to break some long-standing records.

This is joint work with Dolica Akello-Egwel, Klas Markstrom, and Søren Riis.

15/10/2018 4:30 PM

Queens' Building, Room: W316

Yegor Stepanov (QMUL)

Octonions, Albert vectors and the groups of type E_6(F).

We discuss a uniform construction of the groups $\mathrm{E}_6(F)$, where $F$ is any field. In particular, we illuminate some of the subgroup structure of these groups.

22/10/2018 4:30 PM

Queens' Building, Room: W316

Adam Harper (Warwick).

Prime number races with very many competitors.

The prime number race is the competition between different coprime residue classes mod $q$ to contain the most primes, up to a point $x$. Rubinstein and Sarnak showed, assuming two $L$-function conjectures, that as $x$ varies the problem is equivalent to a problem about orderings of certain random variables, having weak correlations coming from number theory. In particular, as $q \rightarrow \infty$ the number of primes in any fixed set of $r$ coprime classes will achieve any given ordering for $\sim1/r!$ values of $x$. In this talk I will try to explain what happens when $r$ is allowed to grow as a function of $q$, concentrating on the lack of uniformity that can arise. This is joint work with Kevin Ford and Youness Lamzouri.

In this talk we review some new results concerning the structure of simple modules (and in particular unitary simple modules) for symmetric groups and their deformations over fields of arbitrary characteristic. If time permits, we will discuss applications in calculating resolutions, (graded) Betti numbers, and CM regularity of certain highly symmetric algebraic varieties.

05/11/2018 4:30 PM

Queens' Building, Room: W316

Andrew Booker (Bristol)

Two results on Artin representations

In 1923, Artin posed a conjecture about the finite-dimensional complex representations of Galois groups of number fields (now called Artin representations). This conjecture, most cases of which are still open, is one of the main motivating problems behind the Langlands programme. After a brief introduction to these topics, I will discuss two recent related results. The first, joint with Min Lee and Andreas Strömbergsson, is a classification of the 2-dimensional Artin representations of small conductor, based on some new explicit versions of the Selberg trace formula. The second extends theorems of Sarnak and Brumley to the effect that certain modular forms with algebraic Fourier coefficients must be associated to Artin representations.

19/11/2018 4:30 PM

Queens' Building, Room: W316

Simon R. Blackburn (Royal Holloway)

The Walnut Digital Signature Algorithm

Walnut is a digital signature algorithm that was first proposed in 2017 by Anshel, Atkins, Goldfeld and Gunnells. The algorithm is based on techniques from braid group theory, and is one of the submissions for the high-profile NIST Post Quantum Cryptography standardisation process. The talk will describe Walnut, and some of the attacks that have been mounted on it. No knowledge of cryptography or the braid group will be assumed. Based on joint work with Ward Beullens (KU Leuven).

03/12/2018 4:30 PM

Queens' Building, Room: W316

Peter J. Cameron (St Andrews)

Permutation groups and regular semigroups

How does the group of units shape the structure of a semigroup? This is a question on which progress was very slow, but the increased knowledge of finite groups resulting from the Classification of Finite Simple Groups has opened new lines of progress. I will talk mainly about the following question. What properties of a permutation group $G$ guarantee that, for all non-permutations $s$, or all in some specified class (say, rank $k$, or given image), the semigroup $\langle G,s\rangle$ is regular, or has some other property of interest?